Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation

TL;DR

This paper investigates the convergence speed of Chernoff approximations for solving heat and transport equations, providing both analytical and numerical insights to support their use as numerical methods for variable coefficient PDEs.

Contribution

The study offers the first detailed analysis of convergence rates of Chernoff approximations for specific PDEs, with graphical demonstrations and implications for numerical methods.

Findings

Chernoff approximations converge to exact solutions for heat and transport equations.

The convergence rate is characterized both analytically and numerically.

Graphical illustrations confirm the theoretical convergence behavior.

Abstract

Paul Chernoff in 1968 proposed his approach to approximations of one-parameter operator semigroups while trying to give a rigorous mathematical meaning to Feynman's path integral formulation of quantum mechanics. In early 2000's Oleg Smolyanov noticed that Chernoff's theorem may be used to obtain approximations to solutions of initial-value problems for linear partial differential equations (LPDEs) of evolution type with variable coefficients, including parabolic equations, Schr\"odinger equation, and some other. Chernoff expressions are explicit formulas containing variable coefficients of LPDE and the initial condition, hence they can be used as a numerical method for solving LPDEs. However, the speed of convergence of such approximations at the present time is understudied which makes it risky to employ this class of numerical methods. In the present paper we take two equations with known solutions (heat equation and transport equation) and study both analytically and numerically the speed of decay of the norm of the difference between Chernoff approximations and exact solutions. We also provide graphical illustrations of convergence and its rate. These model examples, being relatively simple, allow to demonstrate general properties of Chernoff approximations. The observations obtained build a base for the future employment of the approach based on Chernoff's theorem to the problem of construction of new numerical methods for solving initial-value problem for parabolic LPDEs with variable coefficients.